To calculate the average deviation from the mean for the dataset {23, 24, 26, 27, 28, 30, 31}, follow these steps:
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Calculate the Mean: \[ \text{Mean} = \frac{\text{sum of all data points}}{\text{number of data points}} = \frac{23 + 24 + 26 + 27 + 28 + 30 + 31}{7} \]
\[ \text{Mean} = \frac{189}{7} = 27 \]
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Calculate the Deviations from the Mean: \[ \text{Deviations} = |x_i - \text{Mean}| \]
- For 23: \(|23 - 27| = 4\)
- For 24: \(|24 - 27| = 3\)
- For 26: \(|26 - 27| = 1\)
- For 27: \(|27 - 27| = 0\)
- For 28: \(|28 - 27| = 1\)
- For 30: \(|30 - 27| = 3\)
- For 31: \(|31 - 27| = 4\)
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List of Absolute Deviations: The absolute deviations are: {4, 3, 1, 0, 1, 3, 4}.
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Calculate the Average Deviation: \[ \text{Average Deviation} = \frac{\text{sum of absolute deviations}}{\text{number of data points}} = \frac{4 + 3 + 1 + 0 + 1 + 3 + 4}{7} \]
\[ \text{Average Deviation} = \frac{16}{7} \approx 2.2857 \]
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Round the Average Deviation to the Nearest Hundredth: \[ \text{Average Deviation} \approx 2.29 \]
Thus, the average deviation from the mean is 2.29.